Dappy projects his small, wiry Anglo-Greek Cypriot self into an east London photographic studio at a little under a billion miles an hour. "Hello!" he shouts, his face only inches away from mine. "Hello, darling!"

Dino Contostavlos is 23 years old, very loud and, yes, really very small. He's instant. An intense, irrepressible, busy and bossy, inescapably charming springer spaniel of a human. "Hello Dappy," I say. He shakes my hand vigorously and kisses me on the cheek, then bounces off to inspect the far reaches of the studio. The air in the room, which had until that point been calm, composed and perhaps somewhat stagnant, is violently resuscitated by his presence.

"Where is everybody?" he shouts. I shrug; I was hoping he could tell me.

Dappy is an hour-and-a-half late for our interview, but his bandmates – his cousin Tulisa (Tula Contostavlos, 22), and his best friend Fazer (Richard Rawson, 23) – are later still.

Dappy tsks his annoyance. "Where the fuck are they?" He turns to me. He looks distraught. "I'm so sorry, darling! This is so embarrassing!" He runs to the studio door and shouts: "Fazer? Fazer! Let's go, let's go, let's go! T! Ready? Ready, T? Ready? Come on! People are waiting!" A minute passes and then Fazer and Tulisa materialise. They enter the room in a less explosive manner: Fazer (all Gucci and appraising sideways glances) is mid-text. Tulisa (dollish, glossy, watchful) walks gingerly in vertiginous heels.

So there they are: N-Dubz. The hip-hop/R&B/multi-platform pop culture sensation. They've got a devoted fan base, two triple-platinum albums, seven top 20 singles, three sell-out tours, one bestselling book and Being N-Dubz, a massively successful Channel 4 documentary series, under their belt.

In other news: they've experienced not inconsiderable bother with the law, developed a tricky relationship with the tabloid press and clocked up what is, to date, the only celebrity association with the recently banned drug miaow miaow (in April, Dappy was caught on film ingesting methadrone – miaow miaow – in a nightclub, much to the delight of the Sun). That's without touching on a kiss-and-tell story (again, all Dappy's work) and the somewhat difficult coda to an interview on Radio 1's Chris Moyles Show: Dappy took the mobile phone number of an unappreciative texter and began sending her death threats.

Despite which – or, possibly, because of which – when N-Dubz range around me and grin, I am overwhelmed with good feeling towards them.

In a pop scene dominated by the saccharine, the sterile, the X-Factored and the media trained, N-Dubz are a raging, sporadically illegal, unrefined, imperfect and beautiful anomaly. They are authentic, outspoken and raw – and no one saw them coming.

They were formed 11 years ago by Dappy's father, Byron Contostavlos: "Uncle B! Yeah! It was Uncle B who put us together." Contostavlos was a musician (formerly of Mungo Jerry) turned Camden Town barber. He wanted to protect his son, niece and Fazer and inspire them, keep them on the straight and narrow from which they were showing signs of veering, with enthusiasm.

As kids, Dappy, Tulisa and Fazer had an impoverished and harsh upbringing in north London. All three attended Haverstock Hill comprehensive in Chalk Farm, a school that had churned out David and Ed Miliband 15 years earlier. But it is fair to say that N-Dubz's experience of growing up was not especially Milibandesque. Gangs, drugs, criminality and bad behaviour defined their youth. They were shaped by the toughness of their environment, they embraced and revelled in it. They do still, to a degree: the name N-Dubz is a contraction of the Camden postcode, NW1; the bad behaviour remains part of their shtick.

"We were naughty!" says Dappy, with relish. "Me and Fazer, we used to go out on the streets and cause madness."

"Me and Dappy were on the wrong path," confirms Fazer. He pauses, solemnly. He isn't afraid of grand utterances. "Music is the saviour for us."

"We could have went: bam! Prison, stabbed, jail, shot, finished!" adds Dappy. He claps his hands together to emphasise each word.

It took N-Dubz eight years to break through. Uncle B scraped together money with which to hire studio space and pay for three white-label albums. He entered the band into talent competitions: "We won a lot of them!" says Dappy.

"We won all of them!" says Fazer.

"Well, there were only, like, one or two," says Tulisa. She sits quietly; she lets the men talk and chips in only if you ask her something directly or she needs to correct Dappy.

Still, they persevered. They wrote, they gigged, they recorded, they had meetings with record company execs. No one signed them; progress was slow. You were young, I say. That demonstrated a great deal of dedication and discipline.

"All that dedication and discipline was Uncle B," says Dappy. "All of it."

N-Dubz gained a significant fan base while still unsigned and underground; their self-released videos received airplay on urban digital channel Channel U. But in 2007 they began to receive mainstream attention. In November of that year, following exposure on YouTube and spiralling viral support, they were awarded a Mobo for best newcomer. This brought them to the attention of Polydor, which rereleased their single "You Better Not Waste My Time", taking it to No 26 in the charts. Things were looking up.

And then, suddenly, Uncle B died.

I can't imagine how painful that must have been, I say. For all of you, but for Dappy specifically. To lose your mentor, your manager. Your father.

"Yes, but that was when it blew!" says Dappy. His tone doesn't vary; he's still excitable, bouncy, silly, inclined to segue off on wild conversational tangents. "The night after he died, we was on stage, in front of 2,000 people. Crying. Crying on stage!"

Had he been ill for a while?

"He knew he had a problem inside him. He could feel it. He had bronchitis in his lungs. Infect, infect, blocked artery…" Dappy makes a comedy choking noise, squeezes his own throat. "Gone!"

So it was a sudden death as far as you were concerned?

"He died, with the remote control in his hand, watching Channel U."

"Waiting for our video," says Fazer.

"Waiting for our video. I gave him mouth to mouth." Dappy demonstrates.

So, wait. Dappy, you found your father's body?

"Yes! Mouth to mouth, knocked on his head! His head was blue! Cold!"

He stops. I look at the "RIP Dad" tattoo on Dappy's neck. A week later, when I join the band on a video shoot in the south of France, I overhear him telling the video director that he needs to be shot from a certain angle because "it shows the Dad tattoo better". I know that the band's breakthrough album, 2008's Uncle B, and their sixth single, "Papa Can You Hear Me?", were both dedicated to Byron Contostavlos. Dappy's grief is opaque and coded and discordantly intertwined with the earliest stages of N-Dubz's success.

It has been a little over three years since Uncle B died, and a little over two and a half since the band began securing hits and growing their fan base exponentially via Being N-Dubz, the Channel 4 documentary devoted to them. They were dropped by Polydor but soon re-signed to All Around the World, a subsidiary of the same parent company, Universal. No one can tell me exactly how much N-Dubz are currently worth, although Rich Castillo, one of their managers, guesses it's around the £2m mark. "They keep 80% of the money, because they write and produce everything themselves." ("And then they go and spend it all on jewellery," a member of their entourage whispers later.)

Their music has tipped them into ubiquity through the course of 2010. If you think you don't know N-Dubz's songs, you're completely wrong. You do. It's that gimmicky, hyper-commercial blend of R&B and hip-hop with a grimy edge – the one with the contrasting hard and raw lyrical motifs that cover crack deals, Facebook stalking and unwanted pregnancies. It's as buoyant, irrepressible and infectious as Dappy – and it is playing out on any radio station remotely interested in snaring a teenage demographic. You'll know it by Dappy's recurring "Na na, na-ay" and by Tulisa's cod-operatic warblings.

So, I say, you lot are famous!

"Yes!" says Dappy. And: "Wow!"

"Wow!" says Fazer. "Our dreams come true!"

"Thing is," says Tulisa, the voice of pragmatism, "we haven't had time to register it. We've just come back from America, from LA, and stuff, where we're not really known yet, so…"

Yet? You expect that to change?

"Oh, soon we'll be the biggest…" says Dappy.

"Huge," offers Fazer.

"Oh, yeah. We'll be big over there."

"Four hundred per cent," says Fazer. "Four hundred per cent."

Which might even turn out to be the case. N-Dubz are signed to Def Jam in the US, a label that counts Kanye West, Justin Bieber and Rihanna among its artists. In November, the band will release their first single on Def Jam. They're vague on their precise strategy for breaking the US, but as far as I can work out, it revolves around elocution lessons and not using "innit" as a lyric any more. "Or 'You get me'!" says Fazer. "Or 'dahn' when we mean 'down'. Other people around the world, if we say, 'Get dahn!', well, they're going to say, 'What's he talking about?'"

"And also because we've got a vision, a global vision, and we know where we're going in our global vision," says Dappy. "And because me and Tulisa have stopped the arguing and the crap…"

Oh, but hang on: you've stopped the arguing? I say. Dappy and Tulisa's fights are epic, the main narrative strand in the Being N-Dubz documentaries.

"We made a pact," says Tulisa. "Ten years of arguing is enough. We've stopped. It's been a month, now."

And that's it? (I'm disappointed; I'd hoped to witness one in the flesh.)

"I did a Dappy tattoo, as a pact of goodwill," she says. She leans forward and sweeps her super-long hair extensions off her shoulders. The base of her neck reads "Dappy".

"D. A. P. P. Y.," says Fazer, helpfully.

"Now every time she sees I'm being naughty, she goes like…" Dappy tips his head forward. "And I'm like: ooooh!" He emits a high-pitched, fearful scream. Tulisa smiles a satisfied smile.

I am intrigued by Tula Constostavlos. Dappy is preposterously charismatic, and Fazer is a competent sidekick, but Tulisa is a truly interesting proposition, rare by anyone's standards. She looks like the definitive R&B glamour piece: all sparkle, cleavage-encasing cocktail dresses and heavy makeup on a pretty face. I'd initially assumed she was window dressing, but later I'd begun to understand the extent of the power she wields over her bandmates and how central and defining a part of the proceedings she was.

And then I watched Tulisa: My Mum and Me, a BBC3 documentary which revealed that, aged 11, Tulisa had become the primary carer for her mother, Anne, who suffers from an extreme mood disorder, an intense combination of schizophrenia and bipolar disorder.

In the film, Tulisa talked dispassionately about watching her mother being sectioned for the first time when Tulisa was five; about how the stress of the situation had pushed her to self-harm, to eating disorders, to depression. She explained that she had attempted suicide twice before she was 18: she'd tried to overdose on pills at 14 and slashed her wrists three years later. Tulisa began writing songs as therapy at 11; Uncle B and N-Dubz became a refuge for her. When I ask her where she'd be without N-Dubz, she says, steadily: "It would be suicide. I don't want to go on and bring it all back to the depression, but… I would not be here. Let's say that."

We cover vast swaths of ground during our 40-minute chat. Dappy's undisciplined ramblings power us into bizarre territory. We address N-Dubz's experience of celebrity. There are the girls: "I'm not going to lie," says Dappy. "When we first got famous, using our music to get girls and shit. Long time ago. Not any more." No? "No! I've got a wife!" Or rather a long-term, if on-and-off, girlfriend, Kaye Vassell, with whom he has an 18-month-old son. "Yeah. I got one kid, one on the way. Fazer's got five. Ha ha! No! No, he hasn't. He hasn't got any."

And there are the other perks of fame: the discounts. "Twenty-five per cent off a car, just because you're famous!" says Tulisa. "The more money you make, the less you have to pay for things." Which is peculiar? "It is. It is."

"People who can afford stuff and it's free! And those that can't, it's a higher price! It should be the other way round," says Fazer.

We address the fans, who are staggeringly devoted. "I saw one, with a tattoo… of my face! My face! On someone's arm," says Dappy. How was that? "Oh, great! It's absolutely great!"

"I've had the whole of Fulham youth club outside my house for a week," reveals Fazer, obscurely.

And we address the haters, the people who post videos on YouTube: "Like there's this one guy, staring straight at the screen!" says Fazer. "Silence… and then he just goes, 'Fuck N-Dubz! Fuck Dappy! Fuck Fazer! Fuck Tulisa!'" He laughs. Does that bother you? "No. It's like, where we were bought up, if there's a drug dealer that's doing very successful for himself, people are going to hate on him, too."

We then discuss the true extent of Dappy's naughtiness, which, he insists, is talked-up by the tabloid press. "I am a spontaneous guy and you only live once, and…" So how much of it is true? "Half of it." For example: "Like the story where we got kicked out of [a hotel at] Alton Towers." ("N-Dubz Dappy Is Puff Baddie!"The Sun, 30 April 2010.) So it didn't happen? I ask. "No, it did." How? And, why? "Erm… someone was allegedly smoking, in a non-smoking room, and so they asked us to leave." And you accepted that? "Yes, sir! It's a family hotel! But the paparazzi were outside, took the pictures, effed off in the car. We wanted to chase them! But… we didn't."

Well, it probably wouldn't have been the most sensible move to make, I suggest.

"Yes, it would! It would have been very sensible!" says Dappy.

I meet N-Dubz again one week later, at a private airfield in Toulon, where they are filming the video for their new single, "Best Behaviour". This is a lament on the twisted nature of fame, on the flawed logic of embracing the adoration of vast quantities of fans while forsaking a truer relationship with one person. The video mainly involves a lot of mournful disembarking from a private jet, some lonely clattering around a beautiful house high in the hills of Provence and the wielding of a great deal of Louis Vuitton luggage. "When you gonna save me, baby?" sings Dappy, spreading his arms out wide, then crossing them over his chest, while his on-screen love interest dives into the swimming pool behind him and attempts to reach him through the aquarium-like dividing glass wall.

What's the story behind the video? I ask Ben Peters, the director. "Do you want a really deep and conceptual answer to that?" he says. "Or can I just say bling?"

Filming is an interminable process. The hold-ups are myriad and unexplained; N-Dubz endure it with wearied resignation and occasional blow-ups. Dappy's pregnant girlfriend, Kaye, is on set, along with Gino, his son. Dappy plays with Gino; he is uncharacteristically calm in his family's company. But he also appears to have developed a new catchphrase – "We beat the system!" – which he applies to anything from the dodging of an early on-set call time to surviving an ill-advised jump from the roof of a location house into the swimming pool. "Daddy beat the system, GG!" he tells his son, apropos of something or other.

Elsewhere: Tulisa wears a coral kaftan and sleeps between takes; she requires considerable quantities of pizza to be shipped in (she won't eat anything else). Fazer amuses himself quietly in corners and is delighted to discover that Tulisa's buff, video love interest extra can't swim that well.

I talk sartorial reinvention with Dappy and Fazer's stylist, who aches to get both of them into trousers that aren't two sizes too wide and long. I witness Dappy erupting when he discovers there is to be no more food on set that day. "On our last video," he screams, "there was salads and everything!"

He and I end up loitering in an antechamber together; I grab the opportunity to ask him if he's backing either of the Miliband brothers in the Labour leadership election battle.

"Who's that then, darling?" he asks.

David and Ed Miliband?

"Never heard of them, darling."

Oh, but I think you all went to the same school. Not at that same time, but still…

"Who again?"

I google them on my iPhone and show Dappy the image results.

He shakes his head.

"Sorry, darling. I don't know them."

But you're a Labour supporter?

"Very much so, darling! I liked Gordon. I could have done with more Gordon."

Not a fan of Cameron's, then?

"I hate David Cameron," Dappy says, and tells me why in terms so libellous that they can't be printed.

We talk about the future of N-Dubz. Dappy expects that, eventually, he and Tulisa will work independently of each other on solo efforts and that Fazer will write and produce for other artists. He says he has plans for his own fashion line: "Na Na Wear! You're going to get trousers and everything." Is he rich? I ask. "Am I rich? I think I am! A little bit. Money makes it easier and there's no point saying it doesn't." On a personal level, he says that his domestic arrangements – his revitalised relationship with Kaye and her pregnancy – have settled him. "Although… there are all these other pretty girls! I could get into trouble! But I try not to."

I leave N-Dubz in the midst of the private airfield in France. They clamber on and off the jet they've hired for this scene, in line with the director's instructions, hefting the Louis Vuitton luggage, singing the hook to "Best Behaviour". They wave me off and thank me for coming; they are a fantastically courteous bunch of celebrities, perhaps the most civil I've met. "You've got to be polite, in't you darling?" Dappy told me, back in London. "You've got to have manners."

You have, Dappy, I thought at the time. You really have. Now, I think: you wouldn't necessarily turn to N-Dubz first for life lessons. And yet N-Dubz are grafters and also dedicated, loyal, authentic, unspoiled by success, ingenuous, unexpectedly kind, inadvertently funny and, yes, really very polite. All in all, you could do a lot worse.